## Stable stage distribution

The next question we want to ask is: does the spotted knapweed population has a stable stage distribution?

Definition: A stable stage distribution corresponds to having constant proportion of individuals in each stage class throughout time.

Example:

Suppose that we are interested in a population with two stages, juvenile and adult. We know that this population is at its stable stage distribution, and each class grows by a factor of two each year. If at time we know there are 6 juveniles and 4 adults, how many will there be in each class at time ?

Since each class grows by a factor of two, we have

But the proportion of individuals in each class at time , and at time hasn't changed, since

Remember that for large time, we found that , where is the eigenvector corresponding to the dominant eigenvalue . Then, one year later we will have

which means that the population vector becomes proportional to the eigenvector , and grows by a factor of each year. This tells us that determines the stable stage distribution.

Question: What is the eigenvector corresponding to the dominant eigenvalue ?

The eigenvector associated with the dominant eigenvalue is found by solving for . The normalized solution is

Question: What is the stable stage structure for the spotted knapweed?

The stable stage structure is determined by the eigenvector associated with the dominant eignvalue. In order to make it easier to interpret the stable stage structure we normalize the eigenvector so that the sum of its elements is equal to one.

To do this, divide each element of the eigenvector by the sum of the elements of the eigenvector to obtain the stable stage structure

This tells us that once the population reaches the the stable stage strucutre, 85.6% of the population will be seeds, 13.5% will be rosettes, and 0.9% will be flowers.

### Graphing the stage changes

We've analytically determined that the spotted knapweed population will increase, and has a long-term growth rate of 5.22. Also, we've determined that the normalised right eigenvector determined the stable stage structure of the population.

Let's confirm these results graphically.

Enter initial numbers of seeds, rosettes, and flowers, and then press Enter.

First, we'll look at a graph of the number of individuals in each stage over time. Does this graph confirm what we found analytically about the population growth?

Next, let's look at a graph of the proportion of individuals in each stage over time. Since we know there is a stable stage structure, we would expect to see the proportions of individuals in each stage become constant in the long-term. Even though the population is still increasing, the proportion of individuals in each stage class remains constant.

Try a few other initial conditions. Do you always see population growth and a stable stage structure?